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1. Introduction
In any reinforced concrete structure, the best efficiency of the rein-
forcement is attained when it is placed in the principal tensile stress
direction. However, in the case of membranes, this assumption is
rarely satisfied. For each combination of loading and at each point
of the structure, there is a principal tensile direction. Therefore,
there are rare cases in which it is possible to determine a single
position of the reinforcement which would be in its best condition.
Furthermore, structures are usually subdivided into many membranes
elements where the stresses are evaluated. It is constructively inad-
equate that the position of reinforcement to be different for each ele-
ment region. Typically, the reinforcement is placed on the structure
in a pattern which makes the construction easier. This paper only
discusses the cases of orthogonal positioning of the reinforcement,
because it is the most common and constructively simpler. For these
reasons, the direction of reinforcement, in general, does not coincide
with the direction of principal tensile stress in membranes.
Due to the aspects described, the design for ULS, i.e., the quanti-
fication of the reinforcement and verification of compressive stress
in concrete, is not easy. However, this problem has been studied
by many researchers and there are some methods for resolution.
One of the first solutions was provided by Baumann [3] in 1972.
He assumes some hypotheses that make his model one of the
simplest to operate.
The solutions proposed for ULS consider cases in which the re-
inforcement is under tension. An also important issue is how to
design membranes which compressive stress in concrete does not
satisfy the strength limit. Possible solutions to this problem are in-
creasing the strength of concrete, increasing membrane thickness
or adopting reinforcement which resists compressive stress.
The objective of this paper is to obtain criteria to use and to design
membranes in the ULS with orthogonal grid reinforcement with at
least one of the directions of reinforcement submitted to compres-
sion. For this study, the method based on Baumann’s criteria [1]
will be used as a basis. The formulation presented in this paper can
be found with more details in Silva [18].
2. Brief history about membrane design
Researchers have long studied the problem of membrane design.
Nielsen [4] proposed a model based on the cracked membrane
concept, in which the reinforcement resists only axial stress and
the concrete is subjected to compressive stress. Baumann [1], in
1972, was probably the first to develop equations that satisfy both
the equilibrium and the compatibility of the membrane. His model
is based on the premise that there is no shear stress along the
cracks. The solutions reached by Baumann [1] and Nielsen [4] are
the same, but deduced from different models.
Gupta [5] uses Baumann’s model to obtain equations that allow
ULS design. Moreover, he solved the problem of obtaining the
minimum amount of reinforcement necessary and the minimum
compression in concrete.
Vecchio and Collins [6] executed an experiment in which thirty re-
inforced concrete panels, with different amounts of reinforcement
in two directions were subjected to several in-plane loadings.
Fialkow [7] adapts the proposed criteria in ACI 318-77 Building
Code [8] of the American Concrete Institute (ACI) to design linear
elements for membrane elements, considering not only the rein-
forcement axial strength and the compressive strength of con-
crete, but also the shear strength provided by the concrete and
the reinforcement.
Based on experiments by Peter [10] and Vecchio and Collins [6],
Gupta and Akbar [9] present a model in order not to only design
membranes, but also to predict their response when subjected to
a set of loads. Gupta and Akbar [9] divide the response of the
membrane into four distinct stages. At the first, the concrete is un-
cracked and the reinforcement has elastic behavior. At the second,
the concrete is cracked and the reinforcement in both directions
has elastic behavior. At the third, the concrete still cracked and
reinforcement in one direction yields. Finally, at the fourth stage,
concrete is cracked and the reinforcements of both directions yield.
At the first stage, the element has elastic behavior. The last stage
refers to the element in the ultimate limit state, a problem for
which there were already some solutions. Gupta e Akbar [9] pres-
ent solutions that allow predicting the behavior of the membrane
for the intermediate stages. To do so, they use some simplifying
assumptions to the problem such as the non-existence of shear
stress between the cracks. They mention the concept of rotation
of cracks, which consists in the change of the cracks direction as
the load increases.
Vecchio and Collins [2] propose the Modified Compression Field
Theory (MCFT). This model considers the effect of tension-stiff-
ening, presupposes the existence of shear stress in the crack,
but only transmitted by aggregate interlock and also considers
the softening of cracked concrete. Because it is more realistic,
considering more variables, it is more complex, but achieves sat-
isfactory results.
Currently, some researchers published studies on this subject. In
his work, Chen [11] compiles some of the above design methods,
such as the one based on the criteria proposed by Baumann [1],
Nielsen [4] and elaborated by Fialkow [7].
Jazra [12] compares the MCFT with the method based on the
Baumann’s criteria, and presents some formulations for designing
compression reinforcement for membranes.
Pereira [17] uses the equations to design membranes to calculate
shells obtaining stresses from a finite element model.
3. Method based on Baumann’s criteria
The design method based on the Baumann’s criteria is probably
the simplest to use to design membranes. For this reason, it was
chosen as the basis of this study.
The method itself has no solution for the case of adopting com-
pression reinforcement, but it will be used to propose a formulation
and criteria to use this reinforcement.
Jazra [12] compares this method with the MCFT proposed by Vec-
chio and Collins [2] and observes that the design obtained with
Baumann`s criteria results higher stress in concrete. This conclu-
sion was expected, because this formulation adopts some hypoth-
eses assuming this result.
Although adequate for ULS design, it will not obtain the same ef-
ficacy in predicting characteristics to verify the SLS, such as strain
and cracking.
The basic hypotheses are:
1. The cracks are approximately parallel and straight.
2. The tensile strength of concrete is null
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IBRACON Structures and Materials Journal • 2012 • vol. 5 • nº 6
T. F. SILVA | J. C. DELLA BELLA